Which of the following pairs contains only intensive properties?

• (A) Mass, Volume
• (B) Density, Temperature
• (C) Volume, Pressure
• (D) Mass, Density

Which compound undergoes Cannizzaro reaction?

• (A) \( \mathrm{CH_3CHO} \)
• (B) \( \mathrm{C_6H_5CHO} \)
• (C) \( \mathrm{CH_3COCH_3} \)
• (D) \( \mathrm{CH_3CH_2OH} \)

The relation between molar conductivity \( \Lambda_m \), conductivity \( \kappa \), and concentration \( C \) of an electrolyte solution is:

• (A) \( \Lambda_m = \dfrac{\kappa}{1000C} \)
• (B) \( \Lambda_m = \dfrac{C}{\kappa \times 1000} \)
• (C) \( \Lambda_m = \dfrac{\kappa \times 1000}{C} \)
• (D) \( \Lambda_m = \kappa C \)

If the concentration of an electrolyte solution decreases, the molar conductivity \( \Lambda_m \) will:

• (A) Decrease
• (B) Increase
• (C) Remain same
• (D) Become zero

A metal has BCC structure. Atomic radius \(=173\,pm\) and molar mass \(M=56\,g mol^{-1}\). The density of the metal is:

• (A) \(7.2\,g cm^{-3}\)
• (B) \(5.6\,g cm^{-3}\)
• (C) \(8.5\,g cm^{-3}\)
• (D) \(3.2\,g cm^{-3}\)

Evaluate the integral:
\[ \int_{1}^{4}\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{e^x}}\right)dx \]

• (A) \(3 + \frac{2}{\sqrt{e}} - \frac{2}{e^2}\)
• (B) \(2 + \frac{2}{\sqrt{e}} - \frac{2}{e^2}\)
• (C) \(2 + \frac{2}{e} - \frac{2}{e^2}\)
• (D) \(3 + \frac{2}{e} - \frac{2}{e^2}\)

Evaluate the integral:
\[ \int \frac{x+1}{x(1+xe^x)^2}\,dx \]

• (A) \( \dfrac{1}{1+xe^x} + C \)
• (B) \( -\dfrac{1}{1+xe^x} + C \)
• (C) \( \ln(1+xe^x) + C \)
• (D) \( \dfrac{xe^x}{1+xe^x} + C \)

If \( h(x) = \sqrt{4f(x) + 3g(x)} \), \( f(1)=4 \), \( g(1)=3 \), \( f'(1)=3 \), \( g'(1)=4 \), then \( h'(1) \) is equal to:

• (A) \( -\dfrac{5}{12} \)
• (B) \( -\dfrac{12}{7} \)
• (C) \( \dfrac{5}{12} \)
• (D) \( \dfrac{12}{5} \)

If \( y = \sin^{-1}\!\left(\dfrac{5x + 12\sqrt{1-x^2}}{13}\right) \), then \( \dfrac{dy}{dx} \) is equal to:

• (A) \( \dfrac{x}{\sqrt{1-x^2}} \)
• (B) \( \dfrac{2}{\sqrt{1-x^2}} \)
• (C) \( -\dfrac{1}{\sqrt{1-x^2}} \)
• (D) \( -\dfrac{x}{\sqrt{1-x^2}} \)

Evaluate the integral:
\[ \int \frac{4x^2 \cot^{-1}(x^3)}{1+x^6}\,dx \]

(where \(C\) is a constant of integration)

• (A) \( -\dfrac{2}{3}(\cot^{-1}x^3) + C \)
• (B) \( \dfrac{2}{3}(\cot^{-1}x^3) + C \)
• (C) \( -\dfrac{2}{3}(\cot^{-1}x^3)^2 + C \)
• (D) \( \dfrac{2}{3}(\cot^{-1}x^3)^2 + C \)

Evaluate the integral:
\[ \int_{0}^{\frac{\pi}{2}} \frac{dx}{1+(\cot x)^{101}} \]

• (A) \( \dfrac{\pi}{2} \)
• (B) \( \dfrac{\pi}{4} \)
• (C) \( \dfrac{\pi}{8} \)
• (D) \( \pi \)